14496451475BEPBe
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The University of Hong Kong**We aren't endorsed by this school
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MATH 607
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Mathematics
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Dec 17, 2024
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NSS Mathematics in Action (2nd Edition)Exam Practice Section B Questions for topics in Book 5B of NSS Mathematics in Action (2nd Edition)QuestionsBookTopic--4ACh 1Quadratic Equations in One Unknown (I)1 6Ch 2Quadratic Equations in One Unknown (II)7 9Ch 3Functions and Graphs10 Ch 4Equations of Straight Lines1117Ch 5More about Polynomials184BCh 6Exponential Functions19 24Ch 7Logarithmic Functions25 35Ch 8More about Equations36Ch 9Variations37 38Ch 10More about Trigonometry39 – 405ACh 1Basic Properties of Circles41 – 49Ch 2Tangents to Circles50 – 52Ch 3Inequalities53 – 58Ch 4Linear Programming59 – 67Ch 5Applications of Trigonometry in 2-dimensional Problems68 – 75Ch 6 Applications of Trigonometry in 3-dimensional Problems76 – 785BCh 7Equations of Circles79Ch 8Locus80 – 81Ch 9Measures of Dispersion82 – 91Ch 10Permutation and Combination92 – 97Ch 11More about Probability1
NSS Mathematics in Action (2nd Edition)Exam Practice Book 5B76.The coordinates of the centre of the circle Care (1, 3). It is given that thex-axis is a tangent to C.(a)Find the equation of the circle C. (2 marks)(b)L: x+ y– 8 = 0 is a line on the same coordinate plane.(i)Does the line Lintersect with the circle C? Explain your answer.(ii)Find the perpendicular distance between the centre and the line L. Hence, find the minimum distance between the line Land the circle C. (8 marks)77.The equation of the circleCis.(a)Find the centre and the radius of the circle C.(2 marks)(b)The line L: y= x– kcuts the circle Cat Mand N.(i)Express the mid-point of MNin terms of k.(ii)If the length of the chord MNisunits, find k.(8 marks)78.In the figure, ABis a diameter of the circle ABCand OCis a tangent to the circle at C. OABand DACare straight lines such that Ois the orthocentre of△OBD.(a)(i)Prove that O, C, Band Dare concyclic. (ii)Is △OCDan isosceles triangle? Explain your answer.(6 marks)(b)A rectangular coordinate system, with Oas the origin, is introduced in the figure so that the coordinates of Aare (3, 0). It is given that the lengths of OCand ACare 4 units and 2.4 units respectively.(i)Find the coordinates of B.(ii)Hence, or otherwise, find the equation of the circle passing through the points O, C,Band D.(6 marks)79.C:is a quadratic curve on the rectangular coordinate plane, where aandbare 2
NSS Mathematics in Action (2nd Edition)Exam Practice real constants. It is given that the roots of are 2 – iand 2 + i.(a)(i)Find aand b.(ii)Hence, find the coordinates of the vertex of the curve C.(5 marks)(b)Qis a moving point on the same rectangular coordinate plane such that the distance between Q and the vertex of the curve Cis equal to the distance between Q and M(m, 2), where mis a constant. Denote the locus of Qby .(i)Express the equation of in terms of m.(ii)Find msuch that passes through the origin.(6 marks)(Assume that 68%, 95% and 99.7% of the data of a normal distribution lie within one, two and three standard deviations from the mean respectively.)80.The time taken for a group of people to finish a fitness test are normally distributed with a mean of 33 s and a standard deviation of 3.5 s.(a)Find the percentage of people whose finishing time is between 29.5 s and 40 s.(3 marks)(b)If 200 people can finish the test within 26 s, find the number of people in the group.(2 marks)(c)Assume that people who finish the fitness test within 37.5 s will pass the test. If Peter is a person in the group and his standard score is 1.4, can he pass the test? Explain your answer.(3 marks)3
NSS Mathematics in Action (2nd Edition)Exam Practice 81.The stem-and-leaf diagram below shows the distribution of the marks of 30 students in class Ain a test.Marks of 30 students in class AStem (tens)Leaf (units)5962 3 5 6 7 7 87891 1 1 2 2 3 3 3 5 8 90 1 2 4 6 7 7 91 2 5The box-and-whisker diagram below shows the distribution of the marks of 30 students in class Bin the same test.Marks of 30 students in class B(a)(i)Find the median, the range and the inter-quartile range of the marks distribution in class A.(ii)State which of the above marks distributions, class Aor B, is more dispersed. Explain your answer.(5 marks)(b)It is given that the standard deviation of the marks distribution in class Bis 5.8. Find the new standard deviation of the marks distribution in class Bafter each of the following adjustments.(i)The mark of each student is increased by 5.(ii)The mark of each student is increased by 5%. (2 marks)(c)A student is randomly selected from each of class Aand class B. Find the probability that the marks of both students are below 78. (2 marks)82.In a lucky draw, a box contains 900 tickets marked with different 3-digit codes from 100 to 999. Find the number of tickets in the box in which(a)exactly two of the digits of the codes are ‘8’,(2 marks)(b)none of the digits of the codes is ‘8’.(2 marks)44560689278Marks
NSS Mathematics in Action (2nd Edition)Exam Practice 83.Each bit of a binary sequence must either be ‘0’ or ‘1’. For example, ‘01011’ is a 5-bit binary sequence.(a)How many different 9-bit binary sequences are starting with ‘1’ and ending with ‘0’?(2marks)(b)How many different 6-bit binary sequences containing two ‘1’s and four ‘0’s are there?(2 marks)84.There are six S5 classes in a school. Find the number of ways to arrange six S5 students to thesix classes if(a)all of them study in the same class,(1 mark)(b)all of them study in different classes,(2 marks)(c)exactly three of them study in S5A.(2 marks)85.A design house produces a series of shirts, in which 8 colours and 6 sizes are available.(a)How many different types of shirts are there in the series? (1 mark)(b)The manager of a boutique shop wants to merchandise 4 shirts of this series from the design house. Staff Aplans to order 4 shirts in the same colour but of different sizes, while staff Bwants 4 shirts of the same size but in different colours. Which staff has a greater number of selections? Explain your answer.(4 marks)86.There are 9 persons in a small wedding party, including the bride, the groom and 7 guests. (a)The photographer asks all of the people in the party to sit in a row to take a photo. Find the number of arrangements if the bride and the groom must sit next to each another.(2 marks)(b)The bride and the groom now choose any two guests, then the four people sit in a row to take a photo together. Find the number of possible arrangements.(3 marks)87.There are 10 different books on a table. Michael wants to put some / all of the books into 5 different empty drawers. Find the number of ways of putting the books in each of the following cases.(a)Each drawer carries exactly one book.(2 marks)(b)Each drawer carries exactly two books.(4 marks)5
NSS Mathematics in Action (2nd Edition)Exam Practice 88.In a country, the identity card code of each citizen consists of 7 characters, in which the first two characters must be upper case alphabets and the remaining five are numbers from 0 to 9. It is known that, for each citizen, the identity card code is unique and all the characters cannotbe repeated. For example, BU07263 is a possible identity card code of a citizen in the country.(a)Suppose the population of the country will be 20 000 000 in 2020. Are there enough identity card codes for all the citizens in the country at that time? Explain your answer.(3 marks)(b)Ken, Amy and Susan are citizens in the country. It is known that the identity card codes ofKen and Amy are YQ71458 and CW96742 respectively. If the alphabets in Susan’s code do not appear in Ken’s and Amy’s codes, and her code ends with an odd number, how many possible identity card codes are there for Susan? (4 marks)89.In the figure, P1, P2, P3are three distinct points on AB, while P4, P5, P6, P7, P8are five distinct points on BC,and P9, P10, P11, P12are four distinct points on CA.(a)How many distinct line segments can be drawn by joining any two points between P1, P2,…, P12from different sides of ABC?(2 marks)(b)Find the number of distinct PxPyPzcan be formed by joining three points among P1, P2,…, P12if (i)all the vertices come from different sides of ABC,(ii)there are no restrictions.(6 marks)6ABCP4P2P3P1P5P6P7P8P9P11P12P10
NSS Mathematics in Action (2nd Edition)Exam Practice 90.Peter and Victor are two of the twelve members in a basketball team. In a basketball game, fivemembers are starters and the remaining seven members are substitutes.(a)Find the number of selections of starters if(i)both Peter and Victor are selected,(ii)either Peter or Victor is selected.(4 marks)(b)Before the game, all the twelve members of the team have to stand in a row to take a photo. Find the number of arrangements if(i)Peter and Victor do not stand next to each another in the photo,(ii)the leftmost 3 members and the rightmost 3 members in the photo are substitutes of the game.(4 marks)91.Rebecca, Jonathan and other 8 students have a gathering in a meeting room every Monday. The seating plan of the meeting room this Monday is shown below.(a)Find the number of arrangements if Rebecca and Jonathan must sit(i)face-to-face each other at the opposite sides of the table,(ii) on the same side of the table.(6 marks)(b)A change will be made to the seating plan next Monday as shown below.Does each of the results of (a)(i) and (ii) increase, decrease or remain unchanged? Explainyour answers.(4 marks)7
NSS Mathematics in Action (2nd Edition)Exam Practice 92.An examination board conducts a certain professional examination every quarter. Tommy starts to take the examination in the first quarter of 2012 until he passes. The probability that he gets a pass in each attempt is 0.8. (a)Find the probability that Tommy passes the examination in at most 2 attempts.(2 marks)(b)Find the probability that Tommy passes the examination in 2012.(2 marks)93.The figure showsa chessboard in a game. The chessboard includes 7 squares in a row, where a coin is placed at the third square from the left. John and Betty take turn to throw a fair dice once. When 1, 2, 3 or 4 are obtained, the coin will move to the left by one square; when 5 or 6 are obtained, the coin will move to the right by two squares. The game ends after two moves.(a)Find the probabilities that the coin will finally land on (i)square X,(ii)square Y.(4 marks)(b)Suppose John wins the game if the coin finally lands on squareX, while Betty wins the game if the coin finally lands on squareZ. Is it a fair game? Explain your answer.(2 marks)94.The following table shows the values and the corresponding quantities of some stamps in an envelope.Value$0.2$0.6$1$1.4 Quantity861210Two stamps are drawn at random from the envelope without replacement. Find the probability of each of the following.(a)The total value of the two stamps is $1.6.(2 marks)(b)The two stamps are of the same value.(2 marks)(c)The total value of the two stamps is greater than $1.2. (2 marks)95.Six S4 students and eight S5 students are committees of Mathematics club. Now, four committees are selected randomly as representatives. Find the probabilities that (a)therepresentatives include equal numbers of S4 students and S5 students,(2 marks)(b)therepresentatives include more S4 students than S5 students,(2 marks)(c)all the representatives are from the same form. (2 marks)8XYZ
NSS Mathematics in Action (2nd Edition)Exam Practice 96.Timothy, Patrick and seven classmates are members of the relay team. Four of them are randomly selected and arranged to participate in an inter-school 4 100 m relay. Find the probability of each of the following. (a)Both Timothy and Patrick are not selected.(2 marks)(b)Patrick is not selected given that Timothy is not selected.(2 marks)(c)Timothy is arranged as either the first or the last runner. (3 marks)97.The following table shows the number of participants joining a summer leadership camp in S5A, S5B and S5C.Number of boysNumber of girlsS5A1012S5B1816S5C97(a)A participant of the camp is selected at random. Find the probabilities that(i)the participant is a girl,(ii)the participant is a boy in S5A.(iii)the participant is not from S5C given that he is a boy. (3 marks)(b)Two participants of the camp are selected one by one at random. Find the probabilities that(i)both of them are girls from the same class,(ii)both of them are girls given that they are from the same class.(5 marks)9